Ok, so now that we know the terms and have a general outline of what the offices look like we can talk about ways to have the most effect with your Reps.
I've been a Legislative Assistant on the House side and now work for a Senate committee; just in case you're wondering what my particular perspective is.
Particularly in the House, where LA's have to write response letters in addition to their legislative duties, it can be many weeks until letters are read and dealt with. Keep in mind, these offices are relatively small. A typical House office has around 5 Leg staff, 1 LC, 1 Staff Assistant, a couple admin staff, maybe a utility infielder, and a couple interns. Senate staffs are much larger but they handle a lot more constituents and issues. Also note: letters are now nuked on the way in so anything plastic melts (so, no pictures, please) and multi-page inkjet letters often become fused together.
Phones-- Phones ring constantly. Keep in mind the folks answering the phones are young, idealistic, and dramatically underpaid (like high teens, in one of the most expensive cities in the country). They aren't trying to give you the runaround, they probably don't know what the bill is you are worried about and they certainly don't know their boss' position on it. That's the job of the Legislative staff. They will keep track of your views. Side note: just for perspective, our office was receiving calls during the Gonzales debate in mid hundreds per day (state + DC) the tally was like 98% opposed. Depending on the issue and the constituency offices may get numbers in the thousands. They are also answering calls from people making appointments and looking for Leg staff so don't be surprised if they can't stay on and chat.
So should you try to get through to an LA? This brings me to my theme for the diary:
You are lobbying, think like a lobbyist.
Lobbyists (good ones, anyway) don't think "this is what I wan't them to do for me," their approach is "here is how I can make your life easier." By all means make your opinions known but realize your opinion alone can only really be another line on the tally sheet. The LA's task on a given issue is to present it fairly to his boss, analyze to practical, policy, and political ramifications, make a recommendation, and write speeches and letters conveying the Member's chosen position. Try looking at it through this prism and seeing how you can fit in.
Do you have a personal anecdote that shows how this will affect you? These are great, because they can sum up an issue perfectly and can also be fodder for speeches.
Are you an expert in the field? Maybe your unique expertise lets you see through the mumbo-jumbo of the other side and you can help the LA present the issue correctly.
Do you know about some person or information that is not getting out in the MSM? People on the Hill are information junkies. If you really know something they probably don't they'll be interested.
You get the idea.
Don't fit into one of these categories? Consider other means of communicating.
Letters to the editor can be really useful. It is assumed that every letter a paper publishes is representative of some number more that agree with it. They also can prompt the editorial board to weigh-in on your issue. The effect of your letter is thus magnified quite a bit. Tip: The Press people scan the local papers looking for things of interest to the boss and any mentions of the boss. If your letter gets published with a line like "Senator X has always been a staunch advocate for people with this problem so it is a little surprising she hasn't weighed in yet" will get your letter in the daily clips that everyone, including the Member, reads.
Phone in to call-in shows, even shudder CSPAN. People listen to them and it's a great way get memes started. In the House office I worked in we used to leave CSPAN on all the time (it's like watching a car wreck, you just can't turn away) and stuff seeps in.
Go to town-hall forums and listening sessions held back in the district. Remember, when you speak you are not just trying to convince the Member in attendence you are trying to convince the audience. You may make the greatest argument in the world but if the Member thinks it will get him unelected it's a no-go.
Finally, if you just want to chat with an LA about an issue and how it may come up, try calling her when it's recess; they'll have time to talk a bit and are generally happy to talk to constituents.
These are just a few thoughts and this is getting long. I hope they help make people a bit more effective.