I don't think this is possible with SQL Developer (but then I don't use it very often).
The SQL client I am using - SQL Workbench/J - can do this.
There are several ways to export this data.
Generate a proprietary script
It can create a SQL script that uses a special (tool specific) notation to reference an external file, something like:
INSERT INTO images
(name, data)
VALUES
('foobar', {$blobfile='blob_r1_c2.data'});
The above statement can only be executed with SQL Workbench again. It is not compatible with any other SQL client.
Use utl_raw
Another alternative is to use a "blob literal", but due to Oracle's limit on 4000 bytes for a character literal, this only works for really small blob values:
INSERT INTO images
(name, data)
VALUES
('foobar', to_blob(utl_raw.cast_to_raw('......')));
where the character literal for the cast_to_raw call would contain the hex values of the BLOB. As this requires 2 characters per "blob byte", you can't handle BLOBs larger than 2000 bytes with that. But that syntax would work for nearly all Oracle SQL tools (if they can handle scripts with very long lines).
SQL*Loader input file
The third alternative is to export the data into a text file that can be imported using SQL*Loader:
The text file would contain something like this:
NAME DATA
foobar blob_r1_c2.data
Together with the following SQL*Loader control file:
OPTIONS (skip=1)
LOAD DATA CHARACTERSET 'WE8ISO8859P15'
INFILE 'images.txt'
APPEND
INTO TABLE IMAGES
FIELDS TERMINATED BY '\t' TRAILING NULLCOLS
(
NAME,
lob_file_data FILLER,
DATA LOBFILE(lob_file_data) TERMINATED BY EOF
)
This can be loaded using SQL*Loader and is thus doesn't need SQL Workbench to import the data.
More details are in the manual
Edit
As Alex has pointed out in his comment, you can also use a DataPump export - but that requires that you have access to the file system on the server. The above solutions all store the data on the client.