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It's best to backup your site using a relatively small part size (say, 100Mb). After the backup is complete, download the parts to your computer and upload them to the new site. Upload and run Kickstart and perform the restoration on the new hosting account. That's the simplest method, the exact procedure Akeeba Backup and its predecessor, JoomlaPack, has been designed for.

There are more advanced ways to do that (e.g. having Akeeba Backup automatically send the backup archive parts and Kickstart to your new hosting account). However, these are more complicated and I have the feeling you are not confident enough to use them. I'd rather you used to simple method I described at the beginning of my reply as there are less things to configure and less things to go wrong.

If you are confident in what you are doing, you CAN have Akeeba Backup transfer a big site's backup by FTP to the new server. You just need to edit your Configuration page.

First make sure that the "Upload kickstart.php" option is checked. This will upload kickstart.php along with the backup archive, saving you one step.

Then you need to set the post-processing engine to "Upload to Remote FTP". Use the FTP connection information to your new host. Make sure you also select the "upload archive parts immediately" option.

The trick in the process is figuring out the optimal part size for split archives (under the Archiver Engine configuration). If it's too high it will time out and the backup will fail. If it's too low you run the risk of the remote FTP server denying your connections and the backup will fail. I'd start with a part size of 100Mb and if that fails start decreasing it to 50Mb, 20Mb, 10Mb or 5Mb.

Also, make sure that your target server has enough disk space available for both the backup and the extracted site (and its database). This is the one thing most people –sometimes yours truly, I plea guilty– forget when transferring large sites.

So you've already established that 100Mb is too much and causes a timeout error :) You can start with a Part Size for Split Archives of 50Mb and start lowering it until the backup + transfer process completes successfully. I will make a wild guess that the sweet spot will be somewhere between a 10 and 20Mb part size.

You're welcome!