Technically, yes you can do this... But in the end it's going to be WAAAAAAY easier to disconnect the new hard drive and boot back off the old one.
If you're the kind of guy that loves beating his head on a wall though...
First way to try is a free file recovery tool like Recuva. Scan the old drive and see what comes up.
If that doesn't work, the fun starts...
Open up a cmd prompt. Type in:
CD x:\$recycle.bin Where x = the drive letter of the old hard drive
dir /a /o-s /s This should spit out a list of SID folder names, like S-1-5-21-4843873838328 etc, with files below. One of these will be the SID of the account your friend logged on with. Hopefully there aren't multiple accounts with deleted files, because in recycler the filenames are changed to gibberish, so you can't tell what the files were called, only what their extension is.
If you can tell what SID your friends account is, right click beside it and select 'mark' then highlight the string and press enter, this will put the folder name in your clipboard. Should look like this: S-1-5-20-1399022810-3114525621-2436237379-12954.
Type:
CD *space* then right click paste the folder string and press enter
Now create a new folder on your C drive, called something simple like New, so C:\New
Copy the folder address of where you are in cmd prompt now so you don't have to type it..
Type:
copy *paste address from above*\*.* C:\New
should look something like this:
copy X:\$Recycle.Bin\S-1-5-20-1399022810-3114525621-2436237379-12954\*.* C:\New
That will copy the recycled files into the new folder you created on your C drive. The files will all work, they'll just be named in gibberish so you'll have to open them up and rename them.
Really not a practical way to do this if you have a lot of files, but if you're desperate...